Here, the first tone of the sequence is the resting or tonic note,
and the tone in parenthesis is the tone an octave above the
first tone, and has double the frequency.
Although this might look like a bit of meaningless but fancy
footwork since we are talking about nothing more than
cyclic permutations of the same set of tones,
the distinctions should be seen in terms of sequence of
Full tone (T) and semitone (S) intervals.

It is the resting point combined with intervallic structure
that gives a mode its auditory qualities.
One might think that these names were taken over directly
from the Liturgical nomenclature for pretty much the same
structures.
One would be very wrong.

One would think that after having corrected the correspondences
to the liturgical modes, that then the objects and names would
be in accordance with those of the ancient Greeks.
One would be even more wrong.
So, clearly these are assumptions definitely not to be made.

We are very fortunate to have some rather early theoretical
writings by Greeks concerning music.
The earliest is seemingly the "Harmonics" of Aristoxenus
(ca. 330 BCE), and so we know what the theorists say.
Understanding what they mean, however, is an entirely
different matter, since written examples of any music to
which the written theory may pertain is almost nonexistent.

This ignorance of the reality of ancient Greek music goes
so far that we are not quite sure what a Greek mode really was.
We only understand ancient Greek music on the level of pitch tempering.
Trying to understand or retrodict what the music was like
is equivalent to predicting the "Variations and Double
Fugue on a Theme of Bach", by Max Reger, merely from a
knowledge of well tempering and possibly a bit about
chords and classical chromatic harmonic cadential formulas.
It can't be done.

For an interesting and thoughtful essay on the problem
that is based in actual research, see
The Ancient Musical Modes: What Were They?,
together with more links there.
The article is by Ed Friedlander, one of the world's good guys
who is also as completely interesting in other than his chosen
field of pathology as he is in pathology.

From the writings of Aristoxenus, one can deduce that the
ancient Greeks used microtonal embellishments, probably
not unlike those stylistic practices in modern Greek music,
and indeed through out middle eastern cultures.
These, unfortunately, have no notational place in western scoring language.

To the Greeks, all thinking about anything serious
was philosophy, and writings on music theory were therefore
in that culture, philosophy.
Ancient Greek philosophers were in one important respect no different
that philosophers in any other culture or time:
the writings were predominantly prescriptive and proscriptive.
See Plato's "Republic", Hobbes "Leviathan" [shudder],
or any other poison of choice.

Whether or not the theories of Greek music were actually
paid attention to in practice is at very least in question.

It is reasonably clear that all theorizing about music is
fundamentally derived from Pythagoras who understood the
intervallic concepts that we call the octave, fifth, and fourth.
Greek music seems to have been automatically wedded to
language either by recitation or by song, but most popularly
involved also two instruments, the aulos and the kithara.
For some other musical definitions
Eric's Treasure Trove: music
may be helpful.

If ancient Greek music is so heavily dependent on the sound
of language (prosody) then we may learn more about it by
knowing and understanding of the sound of (probably Attic)
ancient Greek.
That is another problem altogether, for scholars of ancient
languages.